Interval graphs with unique maxclique ordering

For the linear-time implementation of case4, observe that each vertexuofGcan occur in at most three of the auxiliary graphs: Suppose to the contrary that there are vertices v₁,v2,v3,v₄ such thatu∈N(v_i)for eachi∈[1, 4], and that case4is reached for each of these vertices. As case2does not apply, there are no containments, so we can assume v⁻₁ < v⁻₂ < v⁻₃ < v⁻₄ and v⁺₁ < v⁺₂ < v⁺₃ < v⁺₄. As case 3 applies neither, it follows that v⁺₁ < v⁻₃ and v⁺₂ < v⁻₄. Now letα(u) = [u⁻,u⁺]_{. As} uis a neighbor of all v_i, we know u⁻≤ v⁺₁ and v⁻₄ ≤ u⁺. But this implies thatα(u)either covers α(v₂)alone or together withα(v₁), contradicting that case4is reached forv₂. Note that case3 is subsumed by case4and is only necessary to obtain the linear time bound.

The following is a consequence of Theorem4.5.5and Lemmas4.5.2and4.5.6.

Corollary 4.5.7. Given a graph G and s: E(G) → N⁺, it is possible in O(nm) time (resp., in logspace) to compute a minimal s-respecting interval representation α: V(G) → I of G, or to detect that none exists. Moreover, for any minimal s-respecting interval representation α^′: V(G)→ I^′ of G, there is a hypergraph isomorphism φfromI toI^′ with φ(

α(v)⁾ = α^′(v) for all v∈V(G).

4.5.4 Interval graphs with unique maxclique ordering

As mentioned before, deciding if a graph has an ℓ-respecting interval representation is ^NP-complete [PS97]. This changes however, if the input graph G is required to have a unique interval ordering of its maxcliques (up to reflection); such a graph is calledUCO for unique clique order. Equivalently, a graph Gis UCO if and only if its bundle hypergraphB(G)admits a unique interval representation (up to reflection). On UCO graphs, even the more general problem DCIG (short for distance constrained interval graph) becomes tractable: Additionally toG, a systemDof difference inequalities of the form x_i−x_j ≤ c is given, where the variables are the extreme points of the intervals (strict inequalities are allowed, too). The problem is to decide ifGhas an interval model that satisfies these inequalities. Pe’er and Shamir show that DCIG for UCO graphs is linear-time equivalent to the problem NegCycle, i.e., deciding if a weighted digraph has a negative cycle [PS97]. Based on the following facts, it follows that this problem is NL-complete.

Fact4.5.8. NegCyclewith polynomially bounded weights is^NL-complete.

Proof. To check if a graphGhas a negative cycle, nondeterministically walk throughG (along edges) for at most|V|steps, i.e., guess a vertexv₀∈V(G), and proceed fromv_i by guessingv_i+1 ∈N⁺(v_i) =^{w ∈ V(G)^⏐⏐(v_i,w)∈_E(G)^}_{. When}v_i+1 = v₀ holds, the walk is not extended further. During the walk, store the start vertex v₀, the current vertexv_i, the numberiof steps taken so far, and the accumulated weight of the traversed edges; this can be done in logarithmic space. Accept if the guessed walk returns to the start vertex and has negative weight, otherwise reject. If G contains a negative cycle v₀, . . . ,v_k,v₀, the computation that follows this cycle accepts. On the other hand, any accepting computation witnesses a negative closed walk, which must contain a negative cycle. Thus NegCycleis in^NL.

To show the hardness, consider the following reduction from the^NL-complete prob-lem s-t-Con to decide if there is a directed path from s to t in a given digraph. An instance ⟨G,s,t⟩ of s-t-Con is reduced to the NegCycle instance ⟨G^′,w⟩, where the digraphG^′ is obtained from Gby introducing the edge(t,s)if it is not yet present, and the weight functionw: E(G^′)→Zis given byw(s,t) =−|V(G)|andw(u,v) =1 for all other edges(u,v)∈E(G^′)\ {(s,t)}. If there is a path fromsto tinG, it has length (and thus weight) at most |V(G)| −1. Combining this path with the edge(t,s)results in a negative cycle in G^′. Conversely, if there is a negative cycle in G^′, it must contain the edge(t,s), as this is the only one with negative weight. Cutting(t,s)out of that cycle results in a path froms totin G.

As an intermediate step in their reductions, Pe’er and Shamir consider the problem DiffIneq [PS97], which asks whether a system of difference inequalities admits an integral solution, i.e., the inequalities have the formx_i−x_j ≤c_k or x_i−x_j <c_k, and the constants on the right hand side are polynomially bounded.

Fact4.5.9. DiffIneqis^NL-complete.

Proof. Pe’er and Shamir prove that the general DiffIneqproblem reduces to its special case where all inequalities are weak [PS97, Lemma3.2], by choosing ϵ < _n¹ _(wheren is the number of variables), replacing each strict inequalityx−y <cbyx−y ≤c−ϵ, and scaling the constants byΘ(n) to restore integrality. Thus we can assume that all inequalities are weak.

As ^NLis closed under complementation [Imm88; Sze88], Fact 4.5.8 implies that the problem NoNegCycleof deciding whether a given weighted digraph doesnotcontain a negative cycle is ^NL-complete as well. It is not hard to see that a NoNegCycle instance⟨G,w⟩is equivalent to the DiffIneqinstance⟨X,A⟩with variablesX = _V(_G) and difference inequalities A = ^{x_i−x_j ≤ w(x_j,x_i)^⏐⏐(x_j,x_i)∈E(G)^}[see e.g. AMO93, pp.103sqq.]. Note that this construction can easily be reversed to obtain a reduction in the opposite direction.

For each walk(v₁, . . . ,v_k)of weight W = _∑i^k₌⁻₁¹w(v_i,v_i+1)in⟨G,w⟩, any assignment σ: X→Zthat satisfies all difference inequalities in Aalso satisfiesσ(v_k)≤σ(v₁) +W;

this can be shown by an easy induction on k. If ⟨G,w⟩ contains a negative cycle (v₁, . . . ,v_k−1,v₁), this impliesσ(v₁)<σ(v₁), contradicting the existence of a valid assign-ment.

On the other hand, if⟨G,w⟩does not have a negative cycle, fix an orderingv₁, . . . ,v_n of V(G)such that whenever there is a directed path fromv_j tov_i fori < j, then there is also a directed path fromv_i to v_j. Such an ordering can be obtained by finding the strongly connected components, contracting each of them to a single vertex, taking a topological ordering of the resulting acyclic digraph, and substituting each repre-sentative vertex with the vertices of the strongly connected component it stands for.

To construct a valid assignment σ: X → Z^{, let σ}(v₁) = 0, and for j > 1 let σ(v_j) = min{

σ(v_i) +w(v_i,v_j)^⏐⏐v_i ∈N⁻(v_j)∧i<j}

, where N⁻(v) =^{v ∈V(G)^⏐⏐(u,v)∈E(G)^}. This immediately satisfies all difference inequalities in Athat correspond to edges ofG that go forward w.r.t. the chosen ordering of V(G), and also the others as there are no negative cycles.

4.5 Constrained interval and intersection lengths Together with Fact4.5.9, the next lemma implies that DCIG for UCO graphs is in^NL. Lemma 4.5.10. The disjunctive reduction from DCIG for UCO graphs to DiffIneqgiven by Pe’er and Shamir [PS97] can be implemented in logspace.

Proof. LetG be the input graph, and letD be the distance constraints for the extreme points of the intervals. The idea for the reduction is to extend D with additional constraints to two systems A and A^′ of difference inequalities, so that any interval representation of G that satisfies D corresponds to a satisfying assignment for one of Aand A^′, and vice versa.

Any interval representation ρ of the bundle hypergraph B(G) induces the partial order ≺_ρ on V(_G)_{that hasu}≺_ρ vif and only ifρ(_Bu)lies completely left ofρ(_Bv)_{. For} v∈V(G), letv⁻andv⁺denote the variables for its start and end point, respectively. Both A and A^′ contain all distance constraints inD. Additionally, for each pair of adjacent vertices ifu andv, both systems contain the constraintsu⁻<v⁺andv⁻<u⁺to ensure that their intervals intersect. Foru ≺_ρ v, the inequalityu⁺< v⁻is added to A, and the inequality v⁺< u⁻ is added to A^′. By assumption, ρ is unique up to reversal, so the result is unique up to exchanging AandA^′.

Clearly, any solution to A(or toA^′) also satisfies Dand specifies the extreme points of an interval representation of G. Conversely, in any interval representation α of G that satisfies D, the maxcliques occur in the order specified by eitherρ orρ⁻¹, and thus α satisfies one of A and A^′ [PS97, Lemma 3.1]. It remains to observe that ρ can be computed in logspace by Theorem 4.2.6; the rest of the reduction is easily possible in logspace.

It remains to show that DCIG for UCO graphs is^NL-hard.

Lemma4.5.11. The reduction fromDiffIneqto the special case of DCIG on UCO graphs where all constraints are on the length of intervals [PS97, Section3.2] can be implemented in logspace.

Proof. The reduction works as follows. LetAbe a system of difference inequalities on the variablesX={x₁, . . . ,x_n}. As argued in the proof of Fact4.5.9, it can be assumed that all inequalities are weak, so write A={x_ji −x_ki ≤c_i|i=1, . . . ,n}. FixC>1+_∑^m_i=1|c_i| and let c^′_i =c_i+ (j_i−k_i)C.

The reduction mapsAto the DCIG instance⟨G,D⟩, whereGis the intersection graph of the interval systemI₁∪ I₂∪ I₃, defined by

I₁=^{a_i⏐

⏐i=0, . . . ,n}

wherea_i =^[i,i+1] I₂=^{b_i/2⏐

⏐i=0, . . . , 2n+1}

whereb_i/2 =^[₂ⁱ,₂ⁱ] I₃=^{w_i⏐

⏐i=0, . . . ,m}

wherew_i =^[k_i,j_i]

ifj_i <k_i

andw_i =^[j_i+¹₄,k_i−¹₄^]otherwise.

For integral i, the constraints b⁺_i −b⁻_i = 0 and b⁺_i+1/2−b_i⁻_+1/2 = 1 are included in D.

For j_i > k_i, the constraint w⁺_i −w_i⁻ ≤ c^′_i is added to D; for j_i < k_i, the constraint w⁺_i −w⁻_i ≥ −c_i^′−ϵwithϵ<1/nis added toD.

All these constructions are easily possible in logspace.

Combining Theorem4.5.5and Corollary4.5.7with the algorithm of Theorem4.2.6that computes canonical interval representations of interval hypergraphs allows to compute canonical(ℓ_,s)_{- and}s-respecting interval representations in logspace. The same can also be done in linear time using e.g. the PQ-tree algorithms of Booth and Lueker [BL76].

Using the fact that^FLis closed under composition, it is easy to generalize the logspace results of this section to the case where the prescribed lengths are rational: Bring all lengths to a common denominator and use the resulting numerators. This trans-formation is possible in logspace as iterated integer multiplication is in ^DLogTime -uniform^TC0 [HAB02].

The bottleneck in the O(nm) time algorithm for computing s-respecting interval representations given by Corollary 4.5.7 is the enumeration of R_s (see Lemma 4.5.2).

Can this also be implemented in linear or at least inO(n²)time?

Does the complexity of computing (ℓ,s)- and s-respecting interval representations increase, if the interval and intersection lengths are restricted only for some vertices?

The techniques of this section are not directly applicable in this case, as the algorithm of Lemma4.5.4relies on the uniqueness of the representation, which is not necessarily preserved in the modified scenario.